Even though we have not presented explicitly in the paper, one can show that all bipartite entanglement of \(\vert \bar{\psi }_1 \rangle \)  , \(\bar{\rho }_2\) , and \(\bar{\rho }_3\)  are zero. This means that
\(\vert \bar{\psi }_1 \rangle \)  and \(\bar{\rho }_2\)  are in the GHZ-class[1]}. We do not exactly understand why the W-type entanglement does not
arise. All the questions will be discussed in the future.
